Strong, random field behavior of the Euclidean Dirac propagator

PHYSICAL REVIEW D 98, 033002 (2018)

Strong, random field behavior of the Euclidean Dirac propagator M. P. Fry University of Dublin, Trinity College, Dublin 2, Ireland (Received 18 May 2018; published 13 August 2018) The one-loop effective action of quantum electrodynamics in four dimensions is shown to be controlled by the Euclidean Dirac propagator G in a background potential. After separating the photon self-energy and photon-photon scattering graphs from the effective action the remainder is known to be the logarithm of an entire function of the electric charge of order 4 under mild regularity assumptions on the potential. This input together with quantum electrodynamics’ (QED) lack of an ultrastable vacuum constrain the strong field behavior of G. It is shown that G vanishes in the strong field limit. The relevance of this result to the decoupling of QED from the remainder of the electroweak model for large amplitude variations of the Maxwell field is discussed. DOI: 10.1103/PhysRevD.98.033002

It may seem surprising that anything more remains to be said about the quantized Euclidean Dirac field propagator G in four dimensions in a background potential Aμ , where Gðx; yÞ ¼ ½−ðp − eAÞ þ m Z ∞ 2 2 × dte−tm hxje−t½ðp−eAÞ þeσF=2 jyi; ð1Þ 0

p − eA is the Euclidean Dirac operator D with antiHermitian γ-matrices, γ †μ ¼ −γ μ , with fγ μ ; γ ν g ¼ −2δμν and σ μν ¼ ½γ μ ; γ ν =ð2iÞ. Excepting the general result (2) below there are still no results for the asymptotic behavior of G for strong fields on R4 , including random ones. Intuition is helped by viewing Fμν in Euclidean space as a time-independent four-dimensional magnetic field. Competition between the diamagnetic ðp − eAÞ2 and paramagnetic eσF=2 terms in the exponentiated Hamiltonian in (1) remains an obstruction to the strong field analysis of G. It is the aim of this paper to demonstrate that G vanishes in the strong field limit for a broad class of potentials. It will be explained below why this is of physical interest. The term strong field in this paper refers to the large amplitude variation of a random potential that occurs in a Euclidean functional integral over Aμ. G’s strong-field behavior has no immediate connection with the Minkowski Dirac propagator. For example, the continuation of the four-dimensional magnetic field in G to the Minkowski

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metric can result in imaginary time electric and magnetic fields. When the continuation results in physical fields then a subset of them may simulate laser pulses. Then G becomes the propagator of a charged particle in such a background that can be relevant to current experiments with extremely high-intensity lasers [1]. A theorem is needed that ensures the continued G also vanishes in the strong-field limit. This would be a nontrivial result, considering how difficult it is to calculate the Euclidean G’s strong field limit as discussed in comment 4 below. The most restrictive bound on G known to the author is that of Vafa and Witten [2]: jhαjGjβij ≤

e−mR pffiffiffiffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi hαjαi hβjβi: m

ð2Þ

Here jαi and jβi are any two states of disjoint support, separated by a minimum distance R. The bound is remarkable for its generality: Aμ can be a random potential with no particular symmetry subject to the constraint that it is regular enough for iD to be self-adjoint on a suitable function space. It establishes that G cannot have unbounded growth in a strong magnetic field. The strong-field asymptotic behavior of G is relevant to the extraction of nonperturbative information from the electroweak model. This renormalizable model with its 17 þ 7 adjustable parameters, including three massive Dirac neutrinos and their mixing, is a complete theory of the electroweak interaction. Perturbation expansions are reliable as long as the coupling constants remain small. Pertinent to this, is there a Landau pole [3] when all charged fermions and bosons are included in the effective QED coupling constant at large scales? How does the renormalization group equation for the Higgs coupling behave when summed [4]? These are some of the questions

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perturbation theory cannot answer. The reader who wishes to avoid further discussion of the electroweak model may proceed to the paragraph beginning above Eq. (9). Whether it can be nonperturbatively quantized depends on the convergence of the unexpanded Euclidean functional integrals over all classical field configurations for the vacuum expectation value of its field operators. Integrating out the fermion degrees of freedom results in an effective action depending on 6 lepton and 3 × 6 quark determinants from the neutral weak-current and 3 þ 3 × 3 determinants from the charged weak-current, including color, that are functionals of the Higgs, Maxwell, Z and W fields [5]. Sense can be made of these ill-defined determinants by regularization, renormalization, and the cancellation of their embedded chiral anomalies. When this is done there remain the functional integrals over the Higgs and gauge fields. The question has been asked whether any of these integrals converge [5]. Assuming that the order of doing the integrals is arbitrary it was decided to integrate over the Maxwell field first. Convergence hinges on the growth of the QED one-loop effective action for large amplitude variations of Fμν and hence Aμ provided QED decouples from the remainder of the electroweak model in this limit [5]. Faster than quadratic growth in Fμν would place in doubt whether any process can be calculated nonperturbatively in the electroweak model that includes dynamical fermions. Decoupling happens when the remainder of the electroweak model’s effective action grows no faster than quadratically with Fμν . To see how decoupling can occur consider the determinants contributed by the neutral weak-current  ln det p þ mi − Qi eA −

g gmi ðgV − gA γ 5 Þ= Zþ H 2 cos θW 2M W



− ln detðp þ mi Þ ¼ ln detð1 − Qi eSAÞ   þ ln det 1 þ G −

g gmi ðgV − gA γ 5 Þ= Zþ H 2 cos θW 2MW

 ; ð3Þ

where G ¼ ðp − Qi eA þ mi Þ−1 , S ¼ ðp þ mi Þ−1 , giV ¼ t3L ðiÞ − 2Qi sin2 θW , giA ¼ t3L ðiÞ, t3L ðiÞ is the weak isospin of fermion i, and Qi is its charge in units of the positron electric charge, e; θW is the weak angle; g ¼ e= sin θW ; mi and M W are the fermion and W mass, respectively. The conventions and notation of [6] are followed here. The determinants contributed by the charged weakcurrent are, for quarks,   1 − þ lndet 1− 2 Gt3L ðiÞ¼−1=2 W ð1− γ 5 ÞGt3L ðiÞ¼1=2 W ð1−γ 5 Þ ; 8g

and for leptons   1 lndet 1− 2 Gt3L ðiÞ¼1=2 W þ ð1− γ 5 ÞGt3L ðiÞ¼−1=2 W − ð1−γ 5 Þ ; 8g ð5Þ where Gt3L ðiÞ

 ¼G−G −

 g gmi i i Gt3L ðiÞ : ðg − gA γ 5 Þ= Zþ 2 cos θW V 2MW ð6Þ

Each determinant in (4) and (5) is for a quark pair or lepton pair belonging to the same family such as ðu; dÞ, ðνe ; eÞ, etc. Note the all-pervasive presence of G in (3)–(6). This has its origin in the factorization of the QED effective action from the electroweak model’s that occurs in (3). It is assumed that quark mixing and neutrino mixing are irrelevant to the behavior of the effective action for the large amplitude variations of the Maxwell field. The first term on the right-hand side of (3) contributes to the QED effective action considered below. The second term in (3) and the determinants in (4) and (5) must be renormalized and their triangle graphs’ chiral anomalies cancelled by summation over fermion families, including color. An example of the cancellation of an anomaly in a triangle graph is given in [5] for γ → W þ W −, including quark mixing. Potential chiral anomalies from box graphs such as AAAγ 5 Z and Aγ 5 Zγ 5 Zγ 5 Z are removed by Euclidean C-invariance. By this we mean there exists a matrix C such that Cγ μ C−1 ¼ γ Tμ . In the representation of the γ-matrices used in [5], Eq. (D7), C ¼ γ 3 γ 1 . These operations are done by expanding in e and g through fourth order giving a residue of terms of not more than OððeFÞ2 Þ. The remaining terms are ultraviolet finite and G-dependent, and so a necessary condition for decoupling is that G vanish for large amplitude variations of Aμ . Therefore, information beyond the Vafa-Witten result (2) is required and is the aim of this paper. Equation (9) and the result (15) below indicate that G also controls QED’s one-loop effective action. There are 3 × 2 þ 3 G s corresponding to the three families of quarks and charged leptons, neglecting color. We now turn to the effective action of QED and its relation to G. As the potentials support a gauge-fixed Gaussian measure μðAÞ on S 0 ðR4 Þ, the space of tempered distributions, they are neither differentiable nor locally square-integrable. They will be smoothed by convoluting them with functions f Λ belonging to SðR4 Þ, the space of functions of rapid decrease:

ð4Þ

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AΛμ ðxÞ ¼

Z

d4 yf Λ ðx − yÞAμ ðyÞ:

ð7Þ

STRONG, RANDOM FIELD BEHAVIOR OF THE … Then AΛμ ∈ C∞ and hence is infinitely differentiable. This smoothing process has the beneficial effect of introducing a gauge invariance preserving ultraviolet cutoff required to Rregulate QED. Thus, from the covariance of μðAÞ, dμðAÞAμ ðxÞAν ðyÞ ¼ Dμν ðx − yÞ, where Dμν ðx − yÞ is the free photon propagator in a fixed gauge, obtain Z

dμðAÞAΛμ ðxÞAΛν ðyÞ ¼ DΛμν ðx − yÞ;

ð8Þ

where the regularizing photon propagator DΛμν has the ˆ μν ðkÞjfˆ Λ ðkÞj2 with fˆ Λ ∈ C∞ Fourier transform D 0 , the space ∞ of C functions with compact support such as fˆΛ ðkÞ ¼ 1, k2 ≤ Λ2 and fˆ Λ ðkÞ ¼ 0, k2 ≥ nΛ2 , n > 1 [5]. The AΛμ replace Aμ everywhere in the functional integrals over Aμ except in the measure μðAÞ. In the following the superscript Λ will be omitted with the understanding that Aμ is now a C∞ function. Only when it encounters the measure does Λ reappear. We will deal with the falloff at infinity of the potentials supported by μðAÞ below. Consider any of QED’s renormalized fermion determinants contributing to its effective action corresponding to a specific quark or charged lepton. They can be defined as [7,8,9] ln detren ¼ Π2 þ Π4 þ ln det5 ð1 − eSAÞ;

PHYS. REV. D 98, 033002 (2018) poles for finite x, such as jx − x0 j−β , β > 0. It also means that Aμ falls off at least as fast as 1=jxj for jxj → ∞ and that Aμ is finite at x ¼ 0. Since SA also belongs to I 5 , det5 may be represented as det5 ¼

  4 X ðeSAÞn : ð10Þ ln det5 ð1 − eSAÞ ¼ Tr lnð1 − eSAÞ þ n n¼1 The four subtractions in the brackets in (10) remove from det5 Π2 and Π4 as well as the tadpole and triangle graphs that are set equal to zero as demanded by C-invariance. The remaining n-point graphs, n ≥ 5, contributing to ln det5 can be obtained by expanding ln det5 in powers of e. The gauge invariance of det5 requires that it depends on Fμν only. The representation (10) for det5 is defined only if SA is a compact operator belonging to I r , r > 4. The trace ideal I r (1 ≤ r < ∞) is defined for those compact operators T with r TrðT † TÞ2 < ∞. This means that the eigenstates of T are complete and square-integrable and that the eigenvalues λn P are discrete and satisfy n ð1=jλn jr Þ < ∞. General properties of I r spaces and the properties of determinants of operators belonging to these spaces may be found in [10,11,12,13]. By a theorem of Seiler and Simon [7,8,9,10,14] SA ∈ I r , r > 4 provided Aμ ∈ ∩r>4 Lr ðR4 Þ, m ≠ 0, thereby validating (10) for this class of potentials. This restriction on Aμ means that it has no branch points or

ð11Þ

where the fen g are the discrete, complex eigenvalues of SA [11,13]. Euclidean C-invariance and the reality of det5 for real e require that these appear as quartets en , ¯en or as complex conjugate pairs. Hence det5 is an even function of e. None of the en are on the real axis when m ≠ 0. Since det5 ð0Þ ¼ 0 for real e ≠ 0. Because SA ∈ I r , P1, det5 >4þϵ r > 4, ð1=je jÞ < ∞, ϵ > 0 so that det5 is an n n entire function of e of order 4 [15]. That is, det5 is analytic in e in the entire complex e-plane with j det5 j < AðδÞ expðKðδÞjej4þϵ Þ for any δ > 0 and A, K positive constants. m PWe nowmrelate det5 to G. Since SA ∈ I r>4 , TrðSAÞ ¼ n ð1=en Þ for m ≥ 5. It is evident from (11) that ln det5 has branch points beginning at jej ¼ je1 j ≡ minfjen jg. Therefore, for jej < je1 j the series for ln det5 obtained from (11) can be rearranged to give

ð9Þ

where S is the free fermion propagator, e is its electric charge, and Π2 and Π4 contain the renormalized photon self-energy and γγ-scattering graphs, respectively. The determinant det5 is defined by [10,11,12,13]

 X  ∞  4 Y e ðe=en Þk 1− exp ; en k n¼1 k¼1

ln det5 ¼ −

    1X e 5 1X e 6 − −  5 n en 6 n en

1 1 ¼ − TrðeSAÞ5 − TrðeSAÞ6 −    : 5 6

ð12Þ

Within its radius of convergence, jej < je1 j, this series can be differentiated term-by-term to give e

∂ ln det5 ¼ −Tr½ðeSAÞ5 þ ðeSAÞ6 þ   : ∂e

ð13Þ

We can now analytically continue ln det5 to all e by summing the series:   ∂ 1 5 e ln det5 ¼ −e Tr SASASASA A : ∂e p − eA þ m

ð14Þ

Using G ¼ S þ eSAG and TrðSAÞ5 ¼ 0 gives the final result e

∂ ln det5 ¼ −e6 Tr½ASASASASASAG: ∂e

ð15Þ

Thus G is an integral part of det5 , and accordingly det5 will constrain it. Let Aμ be scaled by L. Suppose for L → ∞Gðx; yÞ ≠ 0, except for sets of x, y of measure zero, and finite for x ≠ y. From (15) j ln det5 ðeLFÞj ¼ OððeLFÞ6 Þ. If ln det5 > 0 for L → ∞ then such growth on the real e-axis is impossible

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for an entire function of e of order 4 [15]. The only possibility is ln det5 < 0 for L → ∞. Then the effective action (9) would decrease as ln detren ðeLFÞ ∼ − ΓðeLÞ6 , L→∞

where Γ > 0 is a homogeneous function of F of degree 6, thereby establishing the absolute stability of QED. Such an ultrastable QED vacuum is unknown in the literature and contradicts the maximal Oð−ðeLÞ2 lnðeLÞÞ decrease of ln detren: when the L2 ðR4 Þ zero modes of D dominate the effective action [5]. The calculation in [5] does not rely on representation (9). Therefore, we conclude for the broad class of potentials for which det5 is defined that G satisfies for x ≠ y lim Gðx; yjLAÞ ¼ 0:

L→∞

ð16Þ

competition between the two terms of H mentioned at the beginning of this paper. (4) Falloff of G: The question arises as to how G decays for potentials on R4 . As stated earlier this is unknown to the author’s knowledge. For example, the contribution of the zero modes mentioned above to G with e ¼ 1 is Gzero modes ðx; yjLAÞ J X

¼ ð2π 2 mÞ−1

ð2j þ 1Þ

j¼0;1=2;1;…

×

j X

0 0 0 0 2j Dj−jm ðψ; θ; φÞDj −jm ðψ ; θ ; φ Þðrr Þ

m¼−j

× Rj ðrÞRj ðr0 ÞK; Some comments on (16): (1) Gauge invariance of (15): G in (15), (16) is gauge-dependent: Gðx; yjA þ ∂λÞ ¼ exp½ieðλðxÞ − λðyÞÞGðx; yjAÞ. Thus, it appears that (15) cannot be gauge-invariant. However, by rederiving (15) when A → A þ ∂λ G can be expanded in powers of A if S ¼ 1=ðp þ mÞ is replaced with 1=ðp − e= ∂ λ þ mÞ. Then Sðx − yÞ → exp ½ieðλðxÞ − λðyÞÞSðx − yÞ. Combining the five phase factors induced by the change in gauge, Gðx; yjA þ ∂λÞ is multiplied by exp ½−ieðλðxÞ − λðyÞÞ so that the phase factor of G is cancelled. The first A in (15) when shifted contributes a remainder depending on ∂λ that must vanish due to the gauge invariance of det5 . (2) The conclusion that G vanishes for strong fields is based on G being embedded in det5 . For potentials with compact support the loop integral defining det5 is cut off at the boundary. So we can only say that (16) holds in the compact support region. (3) Diamagnetism/paramagnetism: G is constructed from a complete set of eigenstates of the Hamiltonian H ¼ ðp − AÞ2 þ σF=2. It is known that the Oð2Þ × Oð3Þ symmetric potentials Mμν xν aðjxjÞ (M νμ ¼ −M μν , M self- or anti-self dual, a ∼ 1=x2 , jxj → ∞) support an unbounded number of L2 ðR4 Þ zero modes on letting Aμ → LAμ , L → ∞ [16]. Hence, scaling Aμ does not necessarily enhance the kinetic energy ðp − AÞ2 relative to the spin term σF=2. Instead, the spectrum of H can remain at its bottom for arbitrarily large fluctuations of Aμ , thereby putting diamagnetism and paramagnetism on an equal footing. Random fields can form deep magnetic wells where Fμν is near zero and Aμ is large due to its nonlocality, effectively decoupling the particle’s spin from Fμν and enhancing diamagnetism. This and the previous comment on zero modes illustrate the

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ð17Þ

where KT ¼ ð0; 1; 0; 0Þ, Djmm0 is a spherical harmonic on the four-dimensional sphere, J is the largest value of j for which 2j þ 2 < L, and Z Ri ðrÞ ¼

0

∞

4jþ3 −2L

drr

e

Rr 0

ds saðsÞ

−1=2

e−L

Rr 0

ds saðsÞ

:

ð18Þ A reliable estimate of G’s falloff for L → ∞ remains. Continuing with the potentials introduced above, the scattering states are also known [16]. Their contribution to G requires summing an infinite series of angular momentum states with their varying Clebsch-Gordan coefficients, integrating this sum over energy and taking the L → ∞ limit. As this has not been done yet the falloff of G remains unknown even in this highly symmetric case. (5) Growth of random potentials: The potentials on which (15) rests may conflict with the growth of a typical A at infinity. By ascribing a property to a typical A we mean all A with the possible exception of a set μðAÞ-measure zero [see above Eq. (7)]. There is indirect evidence that a typical potential’s growth is jxj2 ðln jxjÞβ , β > 1=2 for jxj → ∞ [17]. The evidence is indirect as the analysis in [17] is for a Gaussian measure whose covariance is that of a free, massive, spin-0 boson. No further work relevant to QED is known to the author. Nevertheless we anticipate that the growth of a typical A will be jxjα ðln jxjÞβ for some α; β > 0. It is generally accepted that the functional integrals for the correlation functions of an interacting field theory have to be calculated in finite volume followed by the removal of the volume cutoffs in the thermodynamic limit. This applies to det5 and detren . in particular, in which case the

STRONG, RANDOM FIELD BEHAVIOR OF THE …

PHYS. REV. D 98, 033002 (2018)

growth of a typical potential at infinity will be cut off, allowing det5 to be defined as above. A possible gauge invariant way to implement the introduction of a volume cutoff in ln detren is discussed in Sec. VII of [5]. The two preceding paragraphs do not invalidate (15) which continues to hold under the assumptions required for its derivation.

In conclusion it has been shown that the strong field behavior of QED’s one-loop effective action and the possible decoupling of QED from the remainder of the electroweak model depend on the propagator of a charged fermion in a strong magnetic field. Its strong-field behavior is found to be constrained by the effective action’s connection with an entire function of e of order 4 and QED’s lack of an ultrastable vacuum, leading to (16).

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[10] B. Simon, Trace Ideals and their Applications, 2nd ed., Mathematical Surveys and Monographs (American Mathematical Society, Providence, RI, 2005), Vol. 120. [11] N. Dunford and J. Schwartz, Linear Operators Part II: Spectral Theory (Interscience, New York, 1963). [12] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, Translations of Mathematical Monographs (American Mathematical Society, Providence, RI, 1969), Vol. 18. [13] B. Simon, Adv. Math. 24, 244 (1977). [14] E. Seiler and B. Simon, Commun. Math. Phys. 95, 257 (1984). [15] B. Ya. Levin, Distributions of Zeros of Entire Functions, Translations of Mathematical Monographs (American Mathematical Society, Providence, RI, 1964), Vol. 5. [16] M. P. Fry, Phys. Rev. D 75, 065002 (2007); 75, 069902(E) (2007). An extended version of this paper appears in arXiv: hep-th/0612218. [17] M. Reed and L. Rosen, Commun. Math. Phys. 36, 123 (1974).

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